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In all the time I've studied the language, I've never heard or seen anybody even hint at whether the principles from a given language (like using “burnt toast” vs. “burning toast”) carry over, or if the language has its rules with how that works. Anyone know?

2 Answers
2

This is called a relative clause, and they are pretty interesting in Japanese. Rules from English do not transfer very well at all.

た-form in relative clauses

There are two ways to interpret the た-form of a verb in a relative clause:

as past

as non-past, if:

the verb has a 'result state',

there is no overt actor,

explaining the state change does not require an actor.

Example: allows non-past (semantically, past = non-past)

焦げたトースト
'toast that burned'
'burned toast'

Let's see if non-past is possible:

✓ 焦げている permits the resultative reading

✓ there is no actor in this sentence

✓ toast becoming burnt is not related to any actor

However, note that semantically, the only way to have "burned toast" is for the toast to get burned some point before that, so the non-past and past interpretation are the same. (Essentially, 'burned toast' == 'toast that burned'.)

Example: allows non-past (semantically, past != non-past)

乾いたハンカチ
'a handkerchief that dried'
'a dry handkerchief'

Let's see if non-past is possible:

✓ 乾いている permits the resultative reading

✓ there is no actor in this sentence

✓ a handkerchief becoming dry is not related to any actor

Here, the past and non-past interpretations are different. 乾いたハンカチ does NOT necessarily need to mean that the handkerchief was once wet and then underwent a drying event. It can simply means it is dry. For example, 新しい乾いたハンカチ 'a new, dry handkerchief' is totally fine semantically.

Example: breaks criteria 1

走った人
'a person that ran'
no non-past

✗ 走っている does not permit any resultative reading

Example: breaks criteria 1

乾かしたハンカチ
'a handkerchief that was dried'
no non-past

✗ 乾かしている does not permit the resultative reading

Example: allows non-past, (semantically, past = non-past)

ゆでた卵
'an egg that was boiled'
'a boiled egg'

✓ ゆでている permits the resultative reading

✓ there is no actor in the sentence

✓ an egg boiling has nothing to do with an actor

Notice: even transitive verbs are fine as long as they meet the criteria!

Example: allows non-past, (semantically, past = non-past)

タイプされた論文
'a paper that was typed'
'a typed paper'

✓ タイプされている allows the resultative reading

✓ there is no actor in this sentence

✓ a paper being typed is not related to any actor

Example: breaks criteria 2

太郎によってタイプされた論文
'a paper that was typed by Tarou'
no non-past

✓ タイプされている allows the resultative reading

✗ here, the actor is explicitly mentioned with によって, so clearly this criteria is broken

Example: breaks criteria 2

ペンキを壁に塗った人
'a person who painted the wall'
no non-past

✓ 塗っている allows the resultative reading

✗ the actor (人) is mentioned.

Example: allows non-past (semantically, past != non-past)

帽子をかぶった人
'a person that put a hat on'
'a person that has a hat on'

This one is interesting and really highlights the difference between a subject and an actor:

✓ かぶっている allows the resultative reading

✓ one may think that 人 is the actor and is thus overt, however this is wrong. Consider 花子が太郎の頭にかぶった帽子 'a hat that Hanako put on Tarou's head'. Here, Tarou is only the experiencer, not the actor. But in 太郎が自分の頭にかぶった帽子 'a hat that Tarou put on his own head', Tarou is the experiencer and actor. In our sentence, it is not required for 人 to be the actor, so this is not an overt mention of the actor

✓ 人 moves from the not-wearing-hat state to the wearing-hat state, you do not need to explicitly mention the person who did the actual hat putting.

Note: things tend to only get this messy for 'wearing' verbs like かぶる and はめる

Example: breaks criteria 3

食べたケーキ
'a cake that was eaten (by someone)'
no non-past

✓ 食べている allows the resultative reading

✓ no explicit mention of an actor

✗ the only difference is that the cake has moved inside the stomach of the actor, so explaining the state change requires mentioning the actor

Example: breaks criteria 3

太郎からもらった本
'a book that somebody got from Tarou'
no non-past

✓ もらっている allows the resultative reading

✓ no explicit mention of an actor

✗ the only difference is that the book has moved from Tarou to the actor, so explaining the state change requires mentioning the actor

There is no non-past interpretation. If we want that interpretation, you need to explicitly use a negated ている form:

ゆでていない卵
'an egg that is not boiled'

Dictionary form in relative clauses

If you use the dictionary form of the verb in a relative clause, it tends to take the future reading, although I believe it can occasionally take the habitual reading as well if the semantics suggest it. In the case of 焦げるトースト, I would translate it as "toast that will get burned".

Depends. Some principles may carry over but some may not.
When you take a principle from one language and apply it to another where it does not work the same way (or does not work at all) you are basically making a pragmatic error. Yep, there is a whole field in linguistics devoted to the subject.

Since English is not my mother tongue, I already did make or will make a few pragmatic errors in this reply, so please bear with me.

When examining Japanese verbs pay enough attention to:

transitivity

grammatical aspect (be super cautious about this one)

tense

Transitivity defines who/what is in some state or who/what is the object of an action.
Tense refers to a point in time where the action/state change occurred or will occur and grammatical aspect defines the "evolution" of the state change or action over time.

I've spotted a nice error (in the positive sense) in the accepted answer, so let's use it as an example:

太郎が乾いたハンカチ （？）

Darius probably unintentionally used "to dry" in its intransitive form which is in this case a grammatical error. In English a single word "dry" could represent an adjective, transitive verb or intransitive verb. In Japanese, however, you have separate verbs for intransitive "dry status change (something dried by itself)" 乾く→乾いた and transitive "dry action (somebody dried something)" 乾かす→乾かした. So the correct form of the example above would be:

太郎が乾かしたハンカチ "the handkerchief that Tarou dried"

Tense in Japanese is often reduced to past/not-past. However, tense coupled with grammatical aspect is another story yet.
Please observe:

遊んでいる子供 "Kids who are playing" (now)

道端に死んでいる猫 "A dead cat by the road" (which died some time ago)

As you can see, both verbs are in a ～ている form but the first refers to an activity occurring now whereas the latter refers to something that happened in past but the results are in effect even now.

So long story short, 焦げたトースト means "burnt toast", 焦げるトースト means "a toast which will burn" and 焦げているトースト means "a toast which burnt in past" but the speaker is implying that the result is visible/has some impact even now.

Now, couple this with a relative clause and I could continue writing for another few hours. I'll go to bed but will post some useful references again tomorrow. Sorry about cutting it here.